MEN OF MATHEMATICS
as a physical mathematician, was in the same class as Xewton,
Gauss, and Einstein in his instinct for what is likely to be of
scientific use in mathematics.
As a sequel to his philosophical studies with Johann Friedrich
Herbart (1776-1841), Riemann came to the conclusion in 1850
(he was then twenty-four) that 'a complete, well-rounded
mathematical theory can be established, which progresses from
the elementary laws for individual points to the processes given
to us in the plenum ("continuously filled space") of reality,
without distinction between gravitation, electricity, magnet-
ism, or thermostatics'. This is probably to be interpreted as
Riemann's rejection of all 'action at a distance' theories in
physical science in favour of field theories. In the latter the
physical properties of the 'space1 surrounding a 'charged
particle*, say, are the object of mathematical investigation.
Riemann at this stage of his career seems to have believed in a
space-filling 'ether', a conception now abandoned. But as will
appear from his epochal work on the foundations of geometry,
he later sought the description and correlation of physical
phenomena in the geometry of the 'space' of human experience.
This is in the current fashion, which rejects an existent, unob-
servable ether as a cumbersome superfluity.
Fascinated by his work in physics, Riemann let his pure
mathematics slide for a while and in the autumn of 1850 joined
the seminar in mathematical physics which had just been
founded by Weber, Ulrich, Stern, and Listing. Physical experi-
ments in this seminar consumed the time that scholarly prudence
would have reserved for the doctoral dissertation in mathe-
matics, which Riemann did not submit till he was twenty-five.
One of the leaders in the seminar, Johann Benedict Listing
(1808-82), may be noted in passing, as he probably influenced
Riemann's thought in what was to be (1857) one of his greatest
achievements, the introduction of topologicalmethods into the
theory of functions of a complex variable.
It will be recalled that Gauss had prophesied that analysis
situs would become one of the most important fields of mathe-
matics, and Riemann, by his inventions in the theory of func-
tions, was to give a partial fulfilment of this prophecy. Although